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A Tour of NTL: NTL Implementation and Portability  </title>
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<h1> 
<p align=center>
A Tour of NTL: NTL Implementation and Portability 
</p>
</h1>

<p> <hr> <p>

NTL is designed to be portable, fast,
and relatively easy to use and extend.

<p>
To make NTL portable, no assembly code is used (well, almost none, see below).
This is highly desirable, as architectures are constantly
changing and evolving, and maintaining assembly
code is quite costly.
By avoiding assembly code, NTL should remain usable,
with virtually no maintenance, for many years.

<p>

<h3>Minimal platform requirements</h3>

When the configuration flags <tt>NTL_CLEAN_INT</tt>
and <tt>NTL_CLEAN_PTR</tt> are both <i>on</i> (this is not the default,
see below),
NTL makes two requirements
of its platform,
neither of which are guaranteed by the <tt>C++</tt> language
definition, but are essentially universal:

<ol>
<li>
<tt>int</tt> and <tt>long</tt> quantities, respectively, 
are represented using a 2's complement
representation whose width is equal to the width of <tt>unsigned int</tt>
and <tt>unsigned long</tt>, respectively.
<li>
Double precision floating point
conforms to the IEEE floating point standard.
</ol>

<p>
NTl makes very conservative requirements of the <tt>C/C++</tt> compiler:
<ul>
<li>
it is assumed that the <tt>C</tt> compiler conforms to the original
ANSI <tt>C</tt> standard,
<li>
it is assumed that the <tt>C++</tt> compiler supports all of the
language features described in the <i>second</i> edition of Stroustrup's book,
minus exceptions, templates, and derived types.
</ul>


<p>

<h3>The <tt>NTL_CLEAN_INT</tt> flag</h3>

<p>

The configuration flag <tt>NTL_CLEAN_INT</tt> 
is currently <i>off</i> by default.

<p>
When this flag is off, NTL makes another requirement of its platform;
namely, that arithmetic operations on the type <tt>long</tt>
do not overflow, but simply "wrap around" modulo the word size.
This behavior is <i>not</i> guaranteed by the <tt>C++</tt> standard,
and yet it is essentially universally implemented.
In fact, most compilers will go out of their way to ensure this behavior,
since it is a very reasonable behavior, and since many programs 
implicitly rely on this behavior.

<p>
Making this "wrap around" assumption can lead to slightly more efficient code
on some platforms.
It seems fairly unlikely that one would ever have to turn the
<tt>NTL_CLEAN_INT</tt> flag <i>on</i>, but it seems a good idea
to make this possible, and at the very least 
to identify and isolate the code that
relies on this assumption.




<p>
Actually, with <tt>NTL_CLEAN_INT</tt> off, it is also assumed
that right shifts of signed integers are consistent,
in the sense that if it is sometimes an arithmetic shift,
then it is always an arithmetic shift (the installation
scripts check if right shift appears to be arithmetic, and if so,
this assumption is made elsewhere). 

<p>
It is hard to imagine that there is a platform existing today
(or in the foreseeable future) where these assumptions
are not meet.
However,
 as of version 5.4 of NTL, all of the most 
performance-critical code now works almost as well
with  <tt>NTL_CLEAN_INT</tt> set as without.
The differences are not very significant (maybe 10%).
Therefore, there is hardly any reason to not set this flag.
Also, note that the only code affected by this flag
is the traditional long integer package (which, if you use
GMP as the primary long integer package, is not involved),
and the single-precision modular multiplication routines
defined in <tt>ZZ.h</tt>.

<p>

<h3>The <tt>NTL_CLEAN_PTR</tt> flag</h3>

<p>

The configuration flag <tt>NTL_CLEAN_PTR</tt> 
is currently <i>off</i> by default.

<p>
When this flag is off, NTL makes another requirement of its platform;
namely, that the address space is "flat", and in particular,
that one can test if an object pointed to by a pointer <tt>p</tt>
is located in a array of objects <tt>v[0..n-1]</tt> by testing
if <tt>p &gt;= v</tt> and <tt>p &lt; v + n</tt>.
The <tt>C++</tt> standard does not guarantee that such a test will
work;   the only way to perform this test in a standard-conforming way
is to iteratively test if <tt>p == v</tt>, <tt>p == v+1</tt>, etc.

<p>
This assumption of a "flat" address space is essentially universally 
valid, and making this assumption leads to some more efficient code.
For this reason, the <tt>NTL_CLEAN_PTR</tt> is <i>off</i> by default,
but one can always turn it on, and in fact, the overall performance
penalty should be negligible for most applications.



<h3>Some floating point issues</h3>


<p>
NTL uses floating point arithmetic in a number of places,
including a number of exact computations, where one might
not expect to see floating point.
Relying on floating point may seem prone to errors,
but with the guarantees provided by the IEEE standard,
one can prove the correctness of the NTL code that uses floating point.

<p>
Briefly, the IEEE floating point standard says that basic arithmetic operations
on doubles should work <i>as if</i> the operation were performed with infinite
precision, and then rounded to <tt>p</tt> bits,
where <tt>p</tt> is the precision (typically, <tt>p = 53</tt>).


<p>
Throughout most of NTL, correctness follows from weaker assumptions,
namely
<p>
<ul>
<li>
basic arithmetic operations and conversion from integral types 
produce results with a <i>relative error</i> of 
<tt>2^{-p + 1}</tt> (assuming no overflow),
<li>
multiplication by powers of 2 produce <i>exact</i> results (assuming no overflow),
<li>
basic arithmetic operations on integers represented as doubles and conversions from integral types
to doubles produce <i>exact</i> results, provided the inputs and outputs
are less than <tt>2^p</tt> in absolute value,
<li>
if <tt>y/2 &lt;= x &lt;= 2y</tt>, then <tt>x-y</tt> is computed exactly.
</ul>
Also, NTL allows the compiler to compute <tt>z = x/y</tt> as 
<tt>t = 1/y</tt>, <tt>z = t*x</tt>.

<p>
It is also generally assumed that the compiler does not
do too much "regrouping" of arithmetic expressions involving
floating point.
Most compilers respect the implied grouping of floating point
computations, and NTL goes out of its way to make its
intentions clear: instead of <tt>x = (a + b) + c</tt>,
if the grouping is truly important, this is written 
as <tt>t = a + b; x = t + c</tt>.
Current standards do not allow, and most implementations will not 
perform, any regrouping of this, e.g., <tt>x = a + (b + c)</tt>,
since in floating point, addition and subtraction are not 
associative.

<p>
Unfortunately, some compilers do not do this correctly,
unless you tell them.
With Intel's C compiler <tt>icc</tt>, for example,
you should compile NTL with the flag <tt>-fp-model source</tt>
to enforce strict adherence to floating point standards.
Also, you should be wary of compiling using an optimization
level higher than the default <tt>-O2</tt> --
this may break some floating point assumptions (and maybe
some other assumptions as well).



<p>
One big problem with the IEEE standard is that it allows intermediate
quantities to be computed in a higher precision than the standard
double precision.
This "looseness" in the standard is a substantial impediment to
creating portable software.
Most platforms today implement the "strict" IEEE standard, with no
excess precision.
One notable exception -- the 800 pound gorilla, so to speak -- 
is the Intel x86.

<p>
NTL goes out of its way to ensure that its code is correct with
both "strict" and "loose" IEEE floating point.
This is achieved in a portable fashion throughout NTL, except
for the <tt>quad_float</tt> module, where some desperate hacks,
including assembly code, may be used
to try to work around problems created by "loose" IEEE floating point
<a href="quad_float.txt">[more details]</a>.
But note that even if the <tt>quad_float</tt> package does not work correctly
because of these problems, the only other routines that are affected
are the <tt>LLL_QP</tt> routines in the <tt>LLL</tt> module -- the
rest of NTL should work fine.



<p>
Mostly, NTL does not 
 require that the IEEE floating point 
special quantities "infinity"
and "not a number" are implemented correctly.
This is certainly the case for core code where
floating point arithmetic is used for exact (but fast)
computations, as the numbers involved never get too big (or small).
However, the behavior of
certain explicit floating point computations
(e.g., the <tt>xdouble</tt> and <tt>quad_float</tt> classes,
and the floating point versions of LLL) will be
much more predictable and reliable if "infinity"
and "not a number" are implemented correctly.


<p>
<h3>Implementing long integer arithmetic</h3>
<p>
There are three basic strategies for implementing long integer arithmetic.

<p>
The <i>default</i> strategy is implemented in the 
<i>traditional long integer arithmetic package</i>.
This package is derived from the LIP package originally developed by
A. K. Lenstra, although it has evolved quite a bit within NTL.
This package uses no assembly code and is very portable.

<p>
The <i>second</i> strategy is to use the Gnu Multi-Precision Package (GMP)
as a <i>supplemental long integer arithmetic package</i>.
In this strategy, the representation of long integers is identical
to that in he traditional long integer package.
This representation is incompatible with the GMP representation,
and on-the-fly conversions are done between the two representations
(only when this is sensible).
This strategy typically yields better performance, but requires
that GMP is installed on your platform.

<p>
The <i>third</i> strategy is to use GMP as the 
<i>primary long integer arithmetic package</i>.
In this strategy, the representation of long integers is in a 
form compatible with GMP.
This strategy typically yields the best performance,
but requires
that GMP is installed on your platform, and also
introduces some minor backward incompatibilities in the programming
interface.

<p>
<a href="tour-gmp.html">Go here</a> for more details on the use
of GMP with NTL.

<p>
<h3>Algorithms</h3>
<p>
NTL makes fairly consistent use of asymptotically fast algorithms.

<p>
Long integer multiplication is implemented using the classical
algorithm, crossing over to Karatsuba for very big numbers.
Long integer division is currently only implemented using
the classical algorithm -- unless you use NTL with GMP (version 3 or later)
as either a supplemental or primary long integer package,
which
employs an algorithm that is about twice as slow as multiplication
for very large numbers.
<p>
Polynomial multiplication and division is carried out
using a combination of the classical algorithm, Karatsuba,
the FFT using small primes, and the FFT using the Schoenhagge-Strassen
approach.
The choice of algorithm depends on the coefficient domain.
<p>
Many algorithms employed throughout NTL are inventions
of the author (<a href="http://www.shoup.net">Victor Shoup</a>) 
and his colleagues 
<a href="http://math-www.uni-paderborn.de/~aggathen/joachim.html">Joachim von zur Gathen</a>
and
<a href="http://www4.ncsu.edu/~kaltofen">Erich Kaltofen</a>,
as well as <a href="mailto:abbott@dima.unige.it">John Abbott</a>
and
<a href="http://www.loria.fr/~zimmerma">Paul Zimmermann</a>.

<p>
<h3>
Some of NTL's imperfections
</h3>
<p>

NTL is not a "perfect" library.
Here are some limitations of NTL that a "perfect" library would not have:
<p>
<ul>
<li>
NTL is neither thread-safe nor re-entrant, and making it so 
would require a fundamental redesign.
<p>

<li>
NTL provides only a very crude form of error handling:
print an error message and abort.
For most NTL users, this is quite sufficient.
The alternative would be to have NTL throw exceptions.
Writing code that handles exceptions correctly is quite difficult.
The easy part is throwing and catching exceptions.
The hard part is writing code <i>through which</i> an exception
can be safely and correctly thrown.
Retrofitting NTL to throw exceptions at this late date
would be quite difficult and error prone, and I do not think
that there is much demand for it.

<p>

<li>
NTL does not release all of its resources.
There are some routines which for efficiency reasons will
allocate some memory and never give it back to the system,
so as to avoid re-allocations on subsequent calls.
The amount of memory "stolen" by NTL in this way is fairly reasonable,
and I have heard no complaints yet about its effects.

</ul>


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